Question

Solve the separable differential equation dtdx=x2+164 and find the particular solution satisfying the initial condition x(0)=9

Answer 1

`(dt)/(dx) = x^2 + (1)/(64) \Rightarrow\ dt = \left(x^2 + (1)/(64)\right)dx \Rightarrow \\ \\ \displaystyle \int 1 dt = \int \left(x^2 + (1)/(64)\right)dx \Rightarrow \\ t = (x^3)/(3) + (x)/(64) + C`

C = 9 because all the x terms go away.


`t = (x^3)/(3) + (x)/(64) + 9`